L3 - Non-stationary Processes

Question 1

What do we can an AR(1) process where θ = 1?

Answer 1

• random walk = unit root
• Variance should be constant and not vary over time but in this case, it does!!!
  
  • it is the sum of all the variances of the error terms in the series (more errors through time will then increase the variance!)

Question 2

Examples of how differencing can make a process stationary?

Answer 2

• Why does differencing make a series stationary?
  
  • As it gives us white noise which is stationary by definition
• Example 1
  
  • As we can see that one of the roots of L is 1, we know it is unit root and therefore I(1), so differencing it once will give us a stationary process, I(0)
• Example 2
  
  • We now have two unit-roots from the factorised characteristic equation so we need to difference the process twice

In both cases, the coefficient on the differenced term is the other root of the characteristic equation

Question 3

What is the definition we a relying on when converting a non-stationary process into a stationary one?

Answer 3

Question 4

What are the four common cases of nonstationary processes?

Answer 4

1. Determinist trend processes
2. Simple random walk
3. Random walk with drift
4. Random walk with drift and deterministic trend

Question 5

What does a determinist trend process look like and how can we make it stationary?

Answer 5

• Its the time variable that makes the expected value non-stationary?

Question 6

What does a simple random walk process look like and how can we make it stationary?

Answer 6

Question 7

What does a random walk with a drift process look like and how can we make it stationary?

Answer 7

Question 8

What does a random walk with drift and deterministic trend process look like and how can we make it stationary?

Answer 8

• Random walk as the coefficient on the lag of x is 1

Question 9

How can we determine the order of integration from any general autoregressive process?

Answer 9

Question 10

How do we know if a transformed non-stationary process is now stationary?

Answer 10

• Look at its correlogram

Question 11

How do we perform a Dickey-Fuller unit root test?

Answer 11

Question 12

What statistical table do we use for the Dickey-Fuller unit root test?

Answer 12

• Reject the Null of unit root if the DF critical value is greater than the test statistic (the one you work out)

Question 13

What is the Augumented Dickey-Fuller unit root test?

Answer 13

Question 14

In finance, what variables tend to be stationary?

Answer 14

Most financial variables are I(1)

• Therefore most series are stationary are:
  
  • first differenced
  • growth rates
  • a ratio of two variables

Question 15

how do you perform an ADF test with a drift and deterministic trend?

Answer 15

• P-value is between 1 and 5% so it may be hard to decide whether we reject or accept depending on what level we use

Question 16

What is one problem with the ADF test?

Answer 16

• One problem of this test is its lack of power, i.e. the inability to reject a false null hypothesis
  
  • Can be even more of a problem when the number of observations or time period is small
• Caused by the fact that the alternative hypothesis includes a wide range of values (some very close to the null)
• Alternative tests: KPSS; Phillips-Perron; Modified ADF test

Question 17

What is the KPSS test?

Answer 17

• Only unit root test where the null hypothesis is Stationary

Question 18

What is the Phillips-Perron test?

Answer 18

Question 19

What is the Modified ADF test?

Answer 19

Question 20

How do we test a time series for more than one unit root?

Answer 20

• Second derivative –> accelerator in the underlying macroeconomic variable

Question 21

How does forcasting change with nonstationary processes with a trending variable?

Answer 21

• forcasted value now rely’s on kα and variance also is multipled by k
  
  • varaince increases the further in the future we wish to forecast

Question 22

How does forcasting change with nonstationary processes when it is trend stationary?

Answer 22

Question 23

example of a series correlogram exhibiting seasonality?

Answer 23

• Also can appear when the time series in graphed
  
  • in some quarters sales may be consistently high whereas in others than may be consistently lower

Question 24

What are the ways in which we can deal with seasonality?

Answer 24

Question 25

In reality, what do we do with the data to adjust for seasonality?

Answer 25

• Take the log of the variable than compute differences –> gives you the growth rate
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